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Based on HARPS-N radial velocities (RVs) and TESS photometry, we present a full characterisation of the planetary system orbiting the late G dwarf TOI-561. After the identification of three transiting candidates by TESS, we discovered two additional external planets from RV analysis. RVs cannot confirm the outer TESS transiting candidate, which would also make the system dynamically unstable. We demonstrate that the two transits initially associated with this candidate are instead due to single transits of the two planets discovered using RVs. The four planets orbiting TOI-561 include an ultra-short period (USP) super-Earth (TOI-561 b) with period $P_{\rm b} = 0.45$ d, mass $M_{\rm b} =1.59 \pm 0.36$ M$_\oplus$ and radius $R_{\rm b}=1.42 \pm 0.07$ R$_\oplus$, and three mini-Neptunes: TOI-561 c, with $P_{\rm c} = 10.78$ d, $M_{\rm c} = 5.40 \pm 0.98$ M$_\oplus$, $R_{\rm c}= 2.88 \pm 0.09$ R$_\oplus$; TOI-561 d, with $P_{\rm d} = 25.6$ d, $M_{\rm d} = 11.9 \pm 1.3$ M$_\oplus$, $R_{\rm d} = 2.53 \pm 0.13$ R$_\oplus$; and TOI-561 e, with $P_{\rm e} = 77.2$ d, $M_{\rm e} = 16.0 \pm 2.3$ M$_\oplus$, $R_{\rm e} = 2.67 \pm 0.11$ R$_\oplus$. Having a density of $3.0 \pm 0.8$ g cm$^{-3}$, TOI-561 b is the lowest density USP planet known to date. Our N-body simulations confirm the stability of the system and predict a strong, anti-correlated, long-term transit time variation signal between planets d and e. The unusual density of the inner super-Earth and the dynamical interactions between the outer planets make TOI-561 an interesting follow-up target.